1. Field of the Invention
The present invention relates to computer-based cryptography systems, and particularly to an elliptic curve-based message authentication code, which relies for its security on the elliptic curve discrete logarithm problem, which is well known in mathematics to be a computationally hard problem.
2. Description of the Related Art
In recent years, the Internet community has experienced explosive and exponential growth. Given the vast and increasing magnitude of this community, both in terms of the number of individual users and web sites, and the sharply reduced costs associated with electronically communicating information, such as e-mail messages and electronic files, between one user and another, as well as between any individual client computer and a web server, electronic communication, rather than more traditional postal mail, is rapidly becoming a medium of choice for communicating information. The Internet, however, is a publicly accessible network, and is thus not secure. The Internet has been, and increasingly continues to be, a target of a wide variety of attacks from various individuals and organizations intent on eavesdropping, intercepting and/or otherwise compromising or even corrupting message traffic flowing on the Internet, or further illicitly penetrating sites connected to the Internet.
Encryption by itself provides no guarantee that an enciphered message cannot or has not been compromised during transmission or storage by a third party. Encryption does not assure integrity due to the fact that an encrypted message could be intercepted and changed, even though it may be, in any instance, practically impossible, to cryptanalyze. In this regard, the third party could intercept, or otherwise improperly access, a ciphertext message, then substitute a predefined illicit ciphertext block(s) which that party, or someone else acting in concert with that party, has specifically devised for a corresponding block(s) in the message. The intruding party could thereafter transmit the resulting message with the substituted ciphertext block(s) to the destination, all without the knowledge of the eventual recipient of the message.
The field of detecting altered communication is not confined to Internet messages. With the burgeoning use of stand-alone personal computers, individuals or businesses often store confidential information within the computer, with a desire to safeguard that information from illicit access and alteration by third parties. Password controlled access, which is commonly used to restrict access to a given computer and/or a specific file stored thereon, provides a certain, but rather rudimentary, form of file protection. Once password protection is circumvented, a third party can access a stored file and then change it, with the owner of the file then being completely oblivious to any such change.
Therefore, a need exists for a cryptographic technique that not only provides an extremely high level of security against cryptanalysis, particularly given the sophistication and power of current and future processing technology, but which is also capable of detecting a change made to a ciphertext message. Such a technique could be applied to (but is not limited in its use) secure file storage or safeguarding messages transmitted over an insecure network.
Systems and methods that provide integrity checks based on a secret key are usually called message authentication codes (MACs). Typically, message authentication codes are used between two parties that share a secret key in order to authenticate information transmitted between these parties. An adversary should be unable (with significant probability) to produce any properly authenticated message for any message that he or she has not yet seen. Typically, a party authenticates a message by appending to it the corresponding MAC. The receiving party then applies a verification procedure on the received message and its message authentication code to decide if the transmitted message is authentic. This may be accomplished by having the receiving party compute his or her own message authentication code and check to see whether the received and generated codes match.
Message authentication code methods are used widely in many applications to provide data integrity and data origin authentication. However, MACs provide weaker guarantees than digital signatures, as they can only be used in a symmetric setting, where the parties trust each other. In other words, MACs do not provide non-repudiation of origin. However, MACs are preferred over digital signatures because they are two to three orders of magnitude faster in implementation, and MAC results are four to sixteen bytes long compared to the forty to one hundred and twenty eight bytes for signatures.
In order to use a MAC, a sender and a receiver need to share a secret key k (a random bit string of nk bits with typical values for nk in the range of 56 to 128). In order to protect a message, the sender computes the MAC corresponding to the message, which is a bit-string of nmac bits, and appends this string to the message (typical values for nmac are between 32 and 64). The MAC is a complex function of every bit of the message and the key. On receipt of the message, the receiver recomputes the MAC and verifies that it corresponds to the transmitted MAC value.
With regard to the security of MAC algorithms, an opponent who tries to deceive the receiver still does not know the secret key. For this analysis, it is assumed that he knows the format of the messages, and the description of the MAC algorithm. His goal is to try to inject a fraudulent message and append a MAC value that will be accepted by the receiver. He can choose one of two attack strategies: a forgery attack or a key recovery attack. The forgery consists of predicting the value of MACk(m) for a message m without initial knowledge of the shared key k. If the adversary can do this for a single message, he is said to be capable of “existential forgery”. If the adversary is able to determine the MAC for a message of his choice, he is said to be capable of “selective forgery”. Practical attacks often require that a forgery is verifiable; i.e., that the forged MAC is known to be correct beforehand with a probability near one.
A key recovery attack consists of finding the key k from a number of message/MAC pairs. Such an attack is more powerful than forgery, since it allows for arbitrary selective forgeries. Ideally, any attack allowing key recovery requires approximately 2nk operations (in this case, nk denotes the bit-length of k). Verification of such an attack requires nk/nmac text-MAC pairs.
These attacks can be further classified according to the type of control an adversary has over the device computing the MAC value. In a chosen-text attack, an adversary may request and receive MACs corresponding to a number of messages of his choice, before completing his attack. For forgery, the forged MAC must be on a message different than any for which a MAC was previously obtained. In an adaptive chosen-text attack, requests may depend on the outcome of previous requests. It should be noted that in certain environments, such as in wholesale banking applications, a chosen message attack is not a very realistic assumption: if an opponent can choose a single text and obtain the corresponding MAC, he can already make a substantial profit. However, it is best to remain cautious and to require resistance against chosen text attacks.
In the following, various attacks on MACs are considered: brute force key searching, guessing of the MAC, a generic forgery attack, and attacks based on cryptanalysis. A brute force key search requires a few known message-MAC pairs (approximately nk/nmac, which is between one and four for most MAC algorithms). It is reasonable to assume that such a small number of message-MAC pairs is available. The opponent tries all the possible keys and checks whether they correspond to the given message-MAC pairs. Unlike the case of confidentiality protection, the opponent can only make use of the key if it is recovered within its active lifetime (which can be reasonably short). On the other hand, a single success during the lifetime of the system might be sufficient. This depends on a cost/benefit analysis; i.e., how much one loses as a consequence of a forgery. The only way to preclude a key search is to choose a sufficiently large key.
A second relatively simple attack is in the form of choosing an arbitrary fraudulent message, and appending a randomly chosen MAC value. Ideally, the probability that this MAC value is correct is equal to ½ nmac, where nmac is the number of bits of the MAC value. This value should be multiplied with the expected profit corresponding to a fraudulent message, which results in the expected value of one trial. Repeated trials can increase this expected value, but in a good implementation, repeated MAC verification errors will result in a security alarm (i.e., the forgery is not verifiable). For most applications nmac is between 32 and 64, which is sufficient to make this attack uneconomical.
A generic forgery attack exploits the fact most MAC algorithms consist of the iteration of a simple compression function. The MAC input message m is padded to a multiple of the block size, and is then divided into t blocks denoted m1 through mt. The MAC involves a compression function f and an n-bit (n≧nmac) chaining variable Hi between stage i−1 and stage i, such that H0=IV; Hi=ƒ(Hi-1,mi), where 1≦i≦t; and MACk(m)=g(Ht). Here, g denotes the output transformation. The secret key may be employed in the IV, in f, and/or in g. For an input pair (m, m′) with MACk (m)=g(Ht) and MACk(m′)=g(Ht′), a collision is said to occur if MACk(m)=MACk(m′). This collision is termed an internal collision if Ht=Ht′, and an external collision if Ht≠Ht′, but g(Ht)=g(Ht′).
One form of general forgery attack applies to all iterated MACs. Its feasibility depends on the bit sizes n of the chaining variable and nmac of the MAC result, the nature of the output transformation g, and the number s of common trailing blocks of the known texts (s≧0). A simple way to preclude this attack is to append a sequence number at the beginning of every message and to make the MAC algorithm stateful. This means that the value of the sequence number is stored to ensure that each sequence number is used only once within the lifetime of the key. While this is not always practical, it has the additional advantage that it prevents replay attacks. To add more security against external collisions, the function g can include some form of additional randomization.
The above attacks assume that no shortcuts exist to break the MAC algorithm (either for forgery or for key recovery). Since most existing MAC algorithms are not based on mathematically known hard problems, it is now becoming increasingly important to have MAC methods that are based on mathematically known hard problems such as integer factorization and discrete logarithm problems.
There are, conventionally, three main approaches for MAC design that are based on: a hash function with a secret key; a block cipher with chaining (CBC-MAC); and a dedicated MAC. Compared to the number of block ciphers and hash functions, relatively few dedicated MAC algorithms have been proposed. The main reason for this is that security of dedicated MAC methods need to be evaluated from scratch in order to assess their robustness. On the other hand, the security of MAC methods that are based on well established primitives such as secure block ciphers or hash functions can be based on the security of these underlying primitives, and its security does not have to be assessed from scratch.
The availability of fast dedicated hash functions (such as MD4 and MD5, for example) has resulted in several proposals for MAC algorithms based on these functions. However, these hash functions are weaker than intended, thus they are currently being replaced by RIPEMD-160 and by SHA-1, even though these hash functions are not based on mathematically known hard problems.
One method of using hash functions for MAC is to use secret prefix and secret suffix methods such that MAC(m)=h(k∥m) and MACk(m)=h(m∥k). However, the first equation allows for extension attacks, and the second equation opens the possibility of off-line attacks.
Another method is the “secret envelope” method, which requires that MACk(m)=h(k1∥m∥k2) (for example, Internet RFC 1828). For this method, a security proof may be performed based on the assumption that the compression function of the hash function is pseudo-random. While this is an interesting result, it should be pointed out that the compression function of most hash functions has not been evaluated with respect to this property. Further, 2n/2 known texts does not allow for a forgery or a key recovery attack. Additionally, MDx-MAC extends the envelope method by also introducing secret key material into every iteration. This makes the pseudo-randomness assumption more plausible. Moreover, it precludes the key recovery attack by extending the keys to complete blocks. HMAC is yet another variant of this methodology, which uses a nested construction (also with padded keys), such that MACk(m)=h(k2∥h(m∥k1)).
HMAC is used for providing message authentication in the Internet Protocol. The security of HMAC is guaranteed if the hash function is collision resistant for a secret value H0, and if the compression function itself is a secure MAC for one block (with the secret key in the Hi input and the message in the mi input). While these assumptions are weaker, it is believed that the latter assumption still requires further validation for existing hash functions. It is clear from the above that none of the current MACs based on hash functions are based on mathematically known relatively hard or difficult cryptographic problems.
Block ciphers are presently the most popular algorithms in use for providing data privacy. Block ciphers with a block size n and a key size k can be viewed as a family of permutations on the set of all n-bit strings, indexed by k-bit long encryption keys and possessing certain properties.
Some of the properties that are typically required of block ciphers are simplicity of construction and security. With regard to security, it is usually assumed that the underlying block cipher is secure and that the key size k is chosen so that an exhaustive key search is computationally infeasible. In practice, there are two issues to be considered with respect to security: (i) for a randomly chosen key k, it appears as a random permutation on the set of n-bit strings to any computationally bounded observer (i.e., one who does not have an unlimited amount of processing power available) who does not know k and who can only see encryption of a certain number of plaintexts x of their choice; and (ii) to achieve a so-called semantic security which is resistant to collision attacks such as birthday and meet-in-the-middle attacks. Such attacks have been proven to reduce an exhaustive key search significantly against block ciphers. In practice, most data units (including any typical file, database record, IP packet, or email message) that require encryption are greater in length than the block size of the chosen cipher. This will require the application of the block cipher function multiple times. The encryption of many plaintext blocks under the same key, or the encryption of plaintexts having identical parts under the same key may leak information about the corresponding plaintext. In certain situations, it is impossible to achieve semantic security. The goal then is to leak the minimum possible amount of information.
A further property is scalability. Obviously, no block cipher can be secure against a computationally unbounded attacker capable of running an exhaustive search for the unknown value of k. Furthermore, the development of faster machines will reduce the time it takes to perform an exhaustive key search. There is always a demand for more secure ciphers. It will be advantageous to develop a block cipher that is scalable so that an increase in security can be achieved by simply changing the length of the key rather than changing the block cipher algorithm itself.
Another property is efficiency. It is obvious that block ciphers are made computationally efficient to encrypt and decrypt to meet the high data rates demands of current applications such as in multimedia. Furthermore, since speed of execution is also important, it is advantageous to have block cipher that can be implemented in parallel. Of further interest is random access. Some modes allow encrypting and decrypting of any given block of the data in an arbitrary message without processing any other portions of the message.
Keying material is also an important factor in block ciphers. Some modes require two independent block cipher keys, which leads to additional key generation operations, a need for extra storage space or extra bits in communication. Additionally, of interest, are counter/IV/nonce requirements. Almost all modes make use of certain additional values together with block cipher key(s). In certain cases, such values must be generated at random or may not be reused with the same block cipher key to achieve the required security goals. Further, pre-processing capability is another important factor in block ciphers
The Data Encryption Standard (DES) is a public standard and is presently the most popular and extensively used system of block encryption. DES was adopted as a federal government standard in the United States in 1977 for the encryption of unclassified information. The rapid developments in computing technology in recent years, in particular the ability to process vast amounts of data at high speed, meant that DES could not withstand the application of brute force in terms of computing power. In the late 1990's, specialized “DES cracker” machines were built that could recover a DES key after a few hours by trying possible key values. As a result, after 21 years of application, the use of DES was discontinued by the United States in 1998.
A new data encryption standard called Advanced Encryption Standard (AES) was launched in 2001 in the United States, and it was officially approved with effect from 26 May 2002. However, AES has no theoretical or technical innovation over its predecessor, DES. The basic concept remains the same and, essentially, all that has changed is that the block size n has been doubled. The AES standard specifies a block size of 128 bits and key sizes of 128, 192 or 256 bits. Although the number of 128-bit key values under AES is about 1021 times greater than the number of 56-bit DES keys, future advances in computer technology may be expected to compromise the new standard in due course. Moreover, the increase in block size may be inconvenient to implement.
Furthermore, AES is not based on known computationally hard problems, such as performing factorization or solving a discrete logarithm problem. It is known that encryption methods that are based on known cryptographic problems are usually stronger than those that are not based on such problems. Also, AES provides a limited degree of varying security, 128-bits, 192-bits and 256-bits; i.e., it not truly scalable. It should noted that to have a cipher with a higher degree of security, the cipher would probably need a completely new algorithm which will make the hardware for AES redundant. As a clear example, the hardware for DES cannot be used efficiently for AES. Also, the hardware of the 192-bits AES cipher is not completely compatible with the hardware of the other two ciphers 128-bits and 256-bits.
There are many ways of encrypting data stream that are longer than a block size, where each is referred to as a “mode of operation”. Two of the standardized modes of operation employing DES are Electronic Code Book (ECB), and Cipher Block Chaining (CBC). It should be noted that the security of a particular mode should in principle be equivalent to the security of the underlying cipher. For this, we need to show that a successful attack on the mode of operation gives us almost an equally successful attack on the underlying cipher.
With regard to the ECB mode, in order to encrypt a message of arbitrary length, the message is split into consecutive n-bit blocks, and each block is encrypted separately. Encryption in ECB mode maps identical blocks in plaintext to identical blocks in ciphertext, which obviously leaks some information about plaintext. Even worse, if a message contains significant redundancy and is sufficiently long, the attacker may get a chance to run statistical analysis on the ciphertext and recover some portions of the plaintext. Thus, in some cases, security provided by ECB is unacceptably weak. ECB may be a good choice if all is need is protection of very short pieces of data or nearly random data. A typical use case for ECB is the protection of randomly generated keys and other security parameters.
With regard to CBC mode, in this mode the exclusive-or (XOR) operation is applied to each plaintext block and the previous ciphertext block, and the result is then encrypted. An n-bit initialization vector IV is used to encrypt the very first block. Unlike ECB, CBC hides patterns in plaintext. In fact, it can be proved that there is a reduction of security of CBC mode to security of the underlying cipher provided that IV is chosen at random. The computational overhead of CBC is just a single XOR operation per block encryption/decryption, so its efficiency is relatively good. Further, CBC provides random read access to encrypted data; i.e., to decrypt the i-th block, we do not need to process any other blocks. However, any change to the i-th message block would require re-encryption of all blocks with indexes greater than i. Thus, CBC does not support random write access to encrypted data.
The most serious drawback of CBC is that it has some inherent theoretical problems. For example, if Mi denotes the i-th plaintext block and Ci denotes the i-th ciphertext block, if one observes in a ciphertext that Ci=Cj, it immediately follows that Mi XOR Mj=Ci-1, XOR Cj-1, where the right-hand side of the equation is known. This is called the “birthday” or matching ciphertext attack. Of course, if the underlying cipher is good in the sense of pseudorandom permutation, and its block size is sufficiently large, the probability of encountering two identical blocks in ciphertext is very low.
Another example of its security weakness is its use of XOR-based encryption. A further drawback of CBC is that its randomization must be synchronized between the sending and the receiving correspondent. CBC uses an initialization vector that must be generated at random. This initialization vector must be synchronized between the sending and receiving correspondent for correct decryption.
From the above, it is clear that the security of encrypting a sequence of message blocks using a block cipher depends on two aspects: the security of the underlying block cipher; and the effectiveness of the randomization used in reducing collision attacks when encrypting a sequence of blocks.
With regard to the security of the underlying block cipher, it is known that encryption methods that are based on computationally hard problems, such as performing factorization or solving a discrete logarithm problem, are usually stronger than those that are not based on such problems. Integer factorization can be formulated as follows: For an integer n that is the product of two primes p and q, the problem is to find the values of p and q given n only. The problem becomes harder for larger primes. The discrete logarithm problem can be formulated as follows: Given a value g and a value y whose value is equal to gk defined over a group, find the value of k. The problem becomes harder for larger groups. Although the applications of integer factorization and discrete logarithm problems in designing block ciphers is known, the resulting ciphers are computationally more demanding than those currently used, such as AES.
With regard to the effectiveness of randomization and semantic security, the one time pad is the only unconditionally semantically secure cipher presently in use. With the one time pad, the sequence of keys does not repeat itself. In other words, it is said to have an infinite cycle. However, since the sending and the receiving correspondents have to generate the same random sequence, the one time pad is impractical because of the long sequence of the non-repeating key. As a consequence, the keys to encrypt and decrypt in all private-key systems, including block ciphers, remain unchanged for every message block, or they are easily derived from each other by inference using identical random number generators at the sending and receiving correspondent. Furthermore, these generators must be initialized to the same starting point at both correspondents to ensure correct encryption and decryption. This is true of all the existing block ciphers, including the RNS encryption and decryption method discussed above.
Many methods have been proposed to construct a pseudo-random number generator or adaptive mechanisms for pseudo-random generation of permutations. Such methods include those based on tables that are used to increase randomization. However, no matter how good the randomization property of the underlying generator, it always has a finite number of states and, hence, the numbers generated by existing generators have a finite cycle where a particular sequence is repeated one cycle after other. Therefore, such block ciphers are vulnerable to collision attacks. Thus, the security of such block ciphers is very much dependant on the randomness of the random number generator. The RNS encryption and decryption method described above is not an exception. As a consequence, one can conclude that semantic insecurity is inherent in all existing block ciphers, but with varying degrees.
It the following, existing ciphers where both the sending and the receiving correspondents have to generate the same random sequence will be referred to as synchronized-randomization ciphers. Synchronized-randomization is achieved under the control of a key or some form of an initialization mechanism. Starting from this initial value, the subsequent keys are easily obtained by some form of a random number generator. Therefore, synchronized-randomization between encryption and decryption is guaranteed as long as identical random number generators are used by both correspondents and as long as the generators at both correspondents are synchronized to start from the same initial state. Thus, no unilateral change in the randomization method is allowed in synchronized-randomization.
With regard to MACs based on block ciphers, the most popular presently used MAC algorithm is the CBC-MAC; it has been adopted by many standardization committees including ANSI and ISO/IEC. It is widely used with DES as the underlying block cipher. CBC-MAC is an iterated MAC, with the following compression function: Hi=Ek(Hi-1⊕mi), where 1≦i≦t. Here, Ek(x) denotes the encryption of x using the nk bit key k with an n-bit block cipher E and H0=0. The MAC is then computed as MACk(m)=g(Ht), where g is the output transformation.
A widely used alternative is to replace the processing of the last block with a two-key triple encryption (with keys k1=k and k2); this is commonly known as the ANSI retail MAC: g(Ht)=Ek1(Dk2(Ht))=Ek1(Dk2(Ek1(mt⊕Ht-1))). Here, D denotes decryption. This mapping requires little overhead, and has the additional advantage that it precludes an exhaustive search against the 56-bit DES key. A second alternative is the use of a derived key k′ (as opposed to a second independent key): g(Ht)=Ek′(Ht)=Ek′(Ek(mt⊕Ht-1)). All of these variants, however, are vulnerable to forgery attack, which requires a single chosen message and approximately 2n/2 known messages (for DES, this corresponds to 232 known messages). For m>n, an additional 2nmac−n 2 chosen messages are required, which makes the attack less realistic. For the ANSI retail MAC, one does not only obtain a forgery, but one can also recover the key in time 3.2nk encryptions, compared to 22nk encryptions for exhaustive search. If DES is used, this implies that key recovery may become feasible. Another key recovery attack needs only a single known text, but requires about 2nk MAC verifications. Moreover, it reduces the effective MAC size from m to min(nmac,nk). The security of the ANSI retail MAC against key recovery attacks can be improved at no cost in performance by introducing a double DES encryption in the first and last iteration. This method of improvement requires: H1=Ek2′(Ek1(m1)) and g(Ht)=Ek2′(Ht). Here, k2′ is derived from k2.
An alternative to CBC-MAC is RIPE-MAC, which adds a feedforward: Hi=Ek(Hi-1⊕mi)⊕mi, where 1≦i≦t. This has the advantage that the round function is harder to invert (even for someone who knows the secret key). An output transformation is needed as well. XOR-MAC is another scheme based on a block cipher. It is a randomized algorithm and its security can again be reduced to that of the block cipher. It has the advantage that it is parallellizable and that small modifications to the message (and to the MAC) can be made at very low cost. The use of random bits helps to improve security, but it has a cost in practical implementations. Further, performance is typically 25% to 50% slower than CBC-MAC.
As noted above, the strength of MAC based on block cipher is dependent on the security of the underlying block cipher. Further, none of the current block ciphers are based on a known cryptographically hard problem.
Methods of adapting discrete-logarithm based algorithms to the setting of elliptic curves are known. However, finding discrete logarithms in this kind of group is particularly difficult. Thus, elliptic curve-based crypto algorithms can be implemented using much smaller numbers than in a finite-field setting of comparable cryptographic strength. Therefore, the use of elliptic curve cryptography is an improvement over finite-field based public-key cryptography.
In practice, an elliptic curve group over a finite field F is formed by choosing a pair of a and b coefficients, which are elements within F. The group consists of a finite set of points P(x,y) that satisfy the elliptic curve equation F(x,y)=y2−x3−ax−b=0, together with a point at infinity, O. The coordinates of the point, x and y, are elements of F represented in N-bit strings. In the following, a point is either written as a capital letter (e.g., point P) or as a pair in terms of the affine coordinates; i.e. (x,y).
The elliptic curve cryptosystem relies upon the difficulty of the elliptic curve discrete logarithm problem (ECDLP) to provide its effectiveness as a cryptosystem. Using multiplicative notation, the problem can be described as: given points B and Q in the group, find a number k such that Bk=Q; where k is the discrete logarithm of Q to the base B. Using additive notation, the problem becomes: given two points B and Q in the group, find a number k such that kB=Q.
In an elliptic curve cryptosystem, the large integer k is kept private and is often referred to as the secret key. The point Q together with the base point B are made public and are referred to as the public key. The security of the system, thus, relies upon the difficulty of deriving the secret k, knowing the public points B and Q. The main factor that determines the security strength of such a system is the size of its underlying finite field. In a real cryptographic application, the underlying field is made so large that it is computationally infeasible to determine k in a straightforward way by computing all the multiples of B until Q is found.
At the heart of elliptic curve geometric arithmetic is scalar multiplication, which computes kB by adding together kcopies of the point B. Scalar multiplication is performed through a combination of point-doubling and point-addition operations. The point-addition operations add two distinct points together and the point-doubling operations add two copies of a point together. To compute, for example, B=(2*(2*(2B)))+2B=Q, it would take three point-doublings and two point-additions.
Addition of two points on an elliptic curve is calculated as follows: when a straight line is drawn through the two points, the straight line intersects the elliptic curve at a third point. The point symmetric to this third intersecting point with respect to the x-axis is defined as a point resulting from the addition. Doubling a point on an elliptic curve is calculated as follows: when a tangent line is drawn at a point on an elliptic curve, the tangent line intersects the elliptic curve at another point. The point symmetric to this intersecting point with respect to the x-axis is defined as a point resulting from the doubling.
Table 1 illustrates the addition rules for adding two points (x1,y1) and (x2,y2); i.e., (x3,y3)=(x1,y1)+(x2,y2):
TABLE 1Summary of Addition Rules: (x3, y3) = (x1, y1) + (x2, y2)General Equationsx3 = m2 − x2 − x1y3 = m (x3 − x1 ) + y1 Point Addition  m  =                    y        2            -              y        1                            x        2            -              x        1             Point Doubling (x3, y3) = 2(x1, y1)  m  =                    3        ⁢                                  ⁢                  x          1          2                    -      a              2      ⁢                          ⁢              y        1             (x2 , y2 ) = −(x1, y1 )(x3, y3 ) = (x1, y1) + (−(x2, y2)) = O(x2, y2) = O(x3, y3) = (x1, y1) + O = (x1, y1)−(x1, y1)= (x1, −y1)
For elliptic curve encryption and decryption, given a message point (xm,ym), a base point (xB,yB), and a given key, k, the cipher point (xC,yC) is obtained using the equation (xC,yC)=(xm,ym)+k(xB,yB).
There are two basics steps in the computation of the above equations. The first step is to find the scalar multiplication of the base point with the key, k(xB,yB). The resulting point is then added to the message point, (xm,ym) to obtain the cipher point. At the receiver, the message point is recovered from the cipher point, which is usually transmitted, along with the shared key and the base point (xm,ym)=(xC,yC)−k(xB,yB).
As noted above, the x-coordinate, xm, is represented as an N-bit string. However, not all of the N-bits are used to carry information about the data of the secret message. Assuming that the number of bits of the x-coordinate, xm, that do not carry data is L, then the extra bits L are used to ensure that message data, when embedded into the x-coordinate, will lead to an xm value which satisfies the elliptic curve equation (1). Typically, if the first guess of xm is not on a curve, then the second or third try will be.
Thus, the number of bits used to carry the bits of the message data is (N−L). If the secret data is a K-bit string, then the number of elliptic curve points needed to encrypt the K-bit data is
      ⌈          K              N        -        L              ⌉    .It is important to note that the y-coordinate, ym, of the message point carries no data bits.
Given a cubic equation in x defined over a finite field, F(p), of the form, t=x3+ax+b, where xεF(p) tεF(p) aεF(p) and bεF(p), then any value of x will lead to a value of tεF(p). It should be noted that t could be either quadratic residue or non-quadratic residue. If t is quadratic residue, it can be written as t=y2, and if t is non-quadratic residue, it can be written as t= αy2 where α is a non-quadratic element of F(p); i.e.,
            α      _        ∉            F      ⁡              (        p        )              .  Thus, equation (5), can be written as αy2=x3+ax+b, where α=1 if t is quadratic residue, and α= α if t is non-quadratic residue.
It should be noted that for a specific coefficient a,bεF(p) that when α=1, the resulting curve is an elliptic curve. However, if α= α, this leads to a twist of the elliptic curve obtained with α=1. Thus, any value of xεF(p) will lead to a point (x,√{square root over (α)}y) that is either on an elliptic curve or its twist. If α=1, the point (x,√{square root over (α)}y) is on the elliptic curve. If α= α, the point (x,√{square root over (α)}y) is on its twist.
Elliptic points can be formulated on a twist of an elliptic curve in the same fashion as they are formulated for elliptic curves. As result, elliptic curve cryptography can also be defined on twists of elliptic curves in the same manner as that described above. Equations for point addition on an elliptic curve or its twist are given in Table 2 below. If α=1, the equations are for point addition on an elliptic curve, and when α= α, the equations are for point addition on its twist.
TABLE 2Summary of Addition Rules on elliptic curves or their twists:(x3, {square root over (α)}y3) = (x1, {square root over (α)}y1) + (x2, {square root over (α)}y2)General Equationsx3 = m2 − x2 − x1{square root over (α)}y3 = m (x3 − x1) + {square root over (α)}y1 Point Addition  m  =            α        ⁢                            y          2                -                  y          1                                      x          2                -                  x          1                     Point Doubling (x3, y3) = 2(x1, y1)  m  =                    3        ⁢                                  ⁢                  x          1          2                    -      a              2      ⁢              α            ⁢              y        1             (x2, {square root over (α)}y2) = −(x1, {square root over (α)}y1)(x3, {square root over (α)}y3) = (x1, {square root over (α)}y1) + (−(x2, {square root over (α)}y2)) = O(x2, {square root over (α)}y2) = O (x3, {square root over (α)}y3) = (x1, {square root over (α)}y1) + O = (x1, {square root over (α)}y1)−(x1, {square root over (α)}y1)= (x1, − {square root over (α)}y1)
Encryption and decryption equation are modified accordingly:(xC,√{square root over (α)}yC)=(xm,√{square root over (α)}ym)+k(xB,√{square root over (α)}yB); and (xm,√{square root over (α)}ym)=(xC,√{square root over (α)}yC)−k(xB,√{square root over (α)}yB).
When α=1, the equations are the cryptography equations over an elliptic curve, and when α= α, they define the cryptography equations over its twist. An attack method referred to as power analysis exists, in which the secret information is decrypted on the basis of leaked information. An attack method in which change in voltage is measured in cryptographic processing using secret information, such as DES (Data Encryption Standard) or the like, such that the process of the cryptographic processing is obtained, and the secret information is inferred on the basis of the obtained process is known. As one of the measures against power analysis attack on elliptic curve cryptosystems, a method using randomized projective coordinates is known. This is a measure against an attack method of observing whether a specific value appears or not in scalar multiplication calculations, and inferring a scalar value from the observed result. By multiplication with a random value, the appearance of such a specific value is prevented from being inferred. In the above-described elliptic curve cryptosystem, attack by power analysis, such as DPA or the like, was not taken into consideration. Therefore, in order to relieve an attack by power analysis, extra calculation has to be carried out using secret information in order to weaken the dependence of the process of the cryptographic processing and the secret information on each other. Thus, time required for the cryptographic processing increases so that cryptographic processing efficiency is lowered.
With the development of information communication networks, cryptographic techniques have been indispensable elements for the concealment or authentication of electronic information. Efficiency in terms of computation time is a necessary consideration, along with the security of the cryptographic techniques. The elliptic curve discrete logarithm problem is so difficult that elliptic curve cryptosystems can make key lengths shorter than that in Rivest-Shamir-Adleman (RSA) cryptosystems, basing their security on the difficulty of factorization into prime factors. Thus, the elliptic curve cryptosystems offer comparatively high-speed cryptographic processing with optimal security. However, the processing speed is not always high enough to satisfy smart cards, for example, which have restricted throughput or servers that have to carry out large volumes of cryptographic processing.
The pair of equations for m in Table 1 are referred to as “slope equations”. Computation of a slope equation in finite fields requires one finite field division. Alternatively, the slope computation can be computed using one finite field inversion and one finite field multiplication. Finite field division and finite field inversion are costly in terms of computational time because they require extensive CPU cycles for the manipulation of two elements of a finite field with a large order. Presently, it is commonly accepted that a point-doubling and a point-addition operation each require one inversion, two multiplications, a square, and several additions. At present, there are techniques to compute finite field division and finite field inversion, and techniques to trade time-intensive inversions for multiplications through performance of the operations in projective coordinates.
In cases where field inversions are significantly more time intensive than multiplication, it is efficient to utilize projective coordinates. An elliptic curve projective point (X,Y,Z) in conventional projective (or homogeneous) coordinates satisfies the homogeneous Weierstrass equation: {tilde over (F)}(X,Y,Z)=Y2Z−X3−aXZ2−bZ3=0, and, when Z≠0, it corresponds to the affine point
      (          x      ,      y        )    =            (                        X          Z                ,                  Y          Z                    )        .  Other projective representations lead to more efficient implementations of the group operation, such as, for example, the Jacobian representations, where the triplets (X,Y,Z) correspond to the affine coordinates
      (          x      ,      y        )    =      (                  X                  Z          2                    ,              Y                  Z          3                      )  whenever Z≠0. This is equivalent to using a Jacobian elliptic curve equation that is of the form {tilde over (F)}(X,Y,Z)=Y2−X3−aXZ4−bZ6=0.
Another commonly used projection is the Chudnovsky-Jacobian coordinate projection. In general terms, the relationship between the affine coordinates and the projection coordinates can be written as
      (          x      ,      y        )    =      (                  X                  Z          i                    ,              Y                  Z          j                      )  where the values of i and j depend on the choice of the projective coordinates. For example, for homogeneous coordinates, i=1 and j=1.
The use of projective coordinates circumvents the need for division in the computation of each point addition and point doubling during the calculation of scalar multiplication. Thus, finite field division can be avoided in the calculation of scalar multiplication,
      k    ⁡          (                                    X            B                                Z            B            i                          ,                              Y            B                                Z            B            j                              )        ,when using projective coordinates.
The last addition for the computation of the cipher point,
      (                            X          C                          Z          C          i                    ,                        Y          C                          Z          C          j                      )    ;i.e., the addition of the two points
            (                                    X            m                                Z            m            i                          ,                              Y            m                                Z            m            j                              )        ⁢                  ⁢    and    ⁢                  ⁢          k      ⁡              (                                            X              B                                      Z              B              i                                ,                                    Y              B                                      Z              B              j                                      )              ,can also be carried out in the chosen projection coordinate:
      (                            X          C                          Z          C          i                    ,                        Y          C                          Z          C          j                      )    =            (                                    X            m                                Z            m            i                          ,                              Y                          m              ⁢                                                          ⁢              1                                            Z            m            j                              )        +                  (                                            X              B                                      Z              B              i                                ,                                    Y              B                                      Z              B              j                                      )            .      It should be noted that Zm=1.
However, one division (or one inversion and one multiplication) must still be carried out in order to calculate
            x      C        =                  x        C                    Z        C        i              ,since only the affine x-coordinate of the cipher point, xC, is sent by the sender.
Thus, the encryption of (N−L) bits of the secret message using elliptic curve encryption requires at least one division when using projective coordinates. Similarly, the decryption of a single message encrypted using elliptic curve cryptography also requires at least one division when using projective coordinates.
As noted above, authentication code methods based on known mathematically difficult problems, such as the discrete logarithm problem, are more secure than those that are not based on such problems. The difficulty of using elliptic curve cryptography to develop authentication codes is in the iterative and non-deterministic method needed to embed a bit string into an elliptic curve point. Further, the iterative embedding methods used in existing elliptic curve cryptography have the additional drawback of the number of iterations needed being different for different bit strings that are being embedded. As a consequence, different encryption times are needed for different blocks of bit strings. Such a data dependant encryption time is not suitable for authentication codes, which require data independent generation time.
Thus, an elliptic curve-based message authentication code system and method solving the aforementioned problems is desired.